Analytic number theory j friedlander et al ( 2006) WW

Lecture Notes in Mathematics Editors: J.-M Morel, Cachan F Takens, Groningen B Teissier, Paris 1891 Fondazione C.I.M.E Firenze C.I.M.E means Centro Internazionale Matematico Estivo, that is, International Mathematical Summer Center Conceived in the early fifties, it was born in 1954 and made welcome by the world mathematical community where it remains in good health and spirit Many mathematicians from all over the world have been involved in a way or another in C.I.M.E.’s activities during the past years So they already know what the C.I.M.E is all about For the benefit of future potential users and co-operators the main purposes and the functioning of the Centre may be summarized as follows: every year, during the summer, Sessions (three or four as a rule) on different themes from pure and applied mathematics are offered by application to mathematicians from all countries Each session is generally based on three or four main courses (24−30 hours over a period of 6-8 working days) held from specialists of international renown, plus a certain number of seminars A C.I.M.E Session, therefore, is neither a Symposium, nor just a School, but maybe a blend of both The aim is that of bringing to the attention of younger researchers the origins, later developments, and perspectives of some branch of live mathematics The topics of the courses are generally of international resonance and the participation of the courses cover the expertise of different countries and continents Such combination, gave an excellent opportunity to young participants to be acquainted with the most advance research in the topics of the courses and the possibility of an interchange with the world famous specialists The full immersion atmosphere of the courses and the daily exchange among participants are a first building brick in the edifice of international collaboration in mathematical research C.I.M.E Director Pietro ZECCA Dipartimento di Energetica “S Stecco” Università di Firenze Via S Marta, 50139 Florence Italy e-mail: zecca@unifi.it C.I.M.E Secretary Elvira MASCOLO Dipartimento di Matematica Università di Firenze viale G.B Morgagni 67/A 50134 Florence Italy e-mail: mascolo@math.unifi.it For more information see CIME’s homepage: http://www.cime.unifi.it CIME’s activity is supported by: – Ministero degli Affari Esteri, Direzione Generale per la Promozione e la Cooperazione, Ufficio V – Ministero dell’Istruzione, Università e Ricerca, Consiglio Nazionale delle Ricerche – E.U under the Training and Mobility of Researchers Programme J.B Friedlander · D.R Heath-Brown H Iwaniec · J Kaczorowski Analytic Number Theory Lectures given at the C.I.M.E Summer School held in Cetraro, Italy, July 11–18, 2002 Editors: A Perelli, C Viola ABC Authors and Editors J.B Friedlander J Kaczorowski Department of Mathematics University of Toronto 40 St George street Toronto, ON M5S 2E4 Canada e-mail: frdlndr@math.toronto.edu Faculty of Mathematics and Computer Science Adam Mickiewicz University ul Umultowska 87 61-614 Poznan Poland e-mail: kjerzy@amu.edu.pl D.R Heath-Brown Alberto Perelli Mathematical Institute University of Oxford 24-29 St Giles Oxford OX1 3LB England e-mail: rhb@maths.ox.ac.uk Dipartimento di Matematica Università di Genova Via Dodecaneso 35 16146 Genova Italy e-mail: perelli@dima.unige.it H Iwaniec Carlo Viola Department of Mathematics Rutgers University 110 Frelinghuysen road Piscataway, NJ 08854 USA e-mail: iwaniec@math.rutgers.edu Dipartimento di Matematica Università di Pisa Largo Pontecorvo 56127 Pisa Italy e-mail: viola@dm.unipi.it Library of Congress Control Number: 2006930414 Mathematics Subject Classification (2000): 11D45, 11G35, 11M06, 11M20, 11M36, 11M41, 11N13, 11N32, 11N35, 14G05 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-36363-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-36363-7 Springer Berlin Heidelberg New York DOI 10.1007/3-540-36363-7 This work is subject to copyright All rights are reserved, 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Heidelberg Printed on acid-free paper SPIN: 11795704 41/SPi 543210 Preface The origins of analytic number theory, i.e of the study of arithmetical problems by analytic methods, can be traced back to Euler’s 1737 proof of the divergence of the series 1/p where p runs through all prime numbers, a simple, yet powerful, combination of arithmetic and analysis One century later, during the years 1837-40, Dirichlet produced a major development in prime number theory by extending Euler’s result to primes p in an arithmetic progression, p ≡ a (mod q) for any coprime integers a and q To this end Dirichlet introduced group characters χ and L-functions, and obtained a key result, the non-vanishing of L(1, χ), through his celebrated formula on the number of equivalence classes of binary quadratic forms with a given discriminant The study of the distribution of prime numbers was deeply transformed in 1859 by the appearance of the famous nine pages long paper by Riemann, ă Uber die Anzahl der Primzahlen unter einer gegebenen Gră osse, where the author introduced the revolutionary ideas of studying the zeta-function ζ(s) = ∞ −s (and hence, implicitly, also the Dirichlet L-functions) as an analytic n function of the complex variable s satisfying a suitable functional equation, and of relating the distribution of prime numbers with the distribution of zeros of ζ(s) Riemann considered it highly probable (“sehr wahrscheinlich”) that the complex zeros of ζ(s) all have real part 12 This still unproved statement is the celebrated Riemann Hypothesis, and the analogue for all Dirichlet Lfunctions is known as the Grand Riemann Hypothesis Several crucial results were obtained in the following decades along the way opened by Riemann, in particular the Prime Number Theorem which had been conjectured by Legendre and Gauss and was proved in 1896 by Hadamard and de la Vall´ee Poussin independently During the twentieth century, research subjects and technical tools of analytic number theory had an astonishing evolution Besides complex function theory and Fourier analysis, which are indispensable instruments in prime number theory since Riemann’s 1859 paper, among the main tools and VI Preface contributions to analytic number theory developed in the course of last century one should mention at least the circle method introduced by Hardy, Littlewood and Ramanujan in the 1920’s, and later improved by Vinogradov and by Kloosterman, as an analytic technique for the study of diophantine equations and of additive problems over primes or over special integer sequences, the sieve methods of Brun and Selberg, subsequently developed by Bombieri, Iwaniec and others, the large sieve introduced by Linnik and substantially modified and improved by Bombieri, the estimations of exponential sums due to Weyl, van der Corput and Vinogradov, and the theory of modular forms and automorphic L-functions The great vitality of the current research in all these areas suggested our proposal for a C.I.M.E session on analytic number theory, which was held at Cetraro (Cosenza, Italy) from July 11 to July 18, 2002 The session consisted of four six-hours courses given by Professors J B Friedlander (Toronto), D R Heath-Brown (Oxford), H Iwaniec (Rutgers) and J Kaczorowski (Pozna´ n) The lectures were attended by fifty-nine participants from several countries, both graduate students and senior mathematicians The expanded lecture notes of the four courses are presented in this volume The main aim of Friedlander’s notes is to introduce the reader to the recent developments of sieve theory leading to prime-producing sieves The first part of the paper contains an account of the classical sieve methods of Brun, Selberg, Bombieri and Iwaniec The second part deals with the outstanding recent achievements of sieve theory, leading to an asymptotic formula for the number of primes in certain thin sequences, such as the values of two-variables polynomials of type x2 + y or x3 + 2y In particular, the author gives an overview of the proof of the asymptotic formula for the number of primes represented by the polynomial x2 + y Such an overview clearly shows the role of bilinear forms, a new basic ingredient in such sieves Heath-Brown’s lectures deal with integer solutions to Diophantine equations of type F (x1 , , xn ) = with absolutely irreducible polynomials F ∈ Z[x1 , , xn ] The main goal here is to count such solutions, and in particular to find bounds for the number of solutions in large regions of type |xi | B The paper begins with several classical examples, with the relevant problems for curves, surfaces and higher dimensional varieties, and with a survey of many results and conjectures The bulk of the paper deals with the proofs of the main theorems where several tools are employed, including results from algebraic geometry and from the geometry of numbers In the final part, applications to power-free values of polynomials and to sums of powers are given The main focus of Iwaniec’s paper is on the exceptional Dirichlet character It is well known that exceptional characters and exceptional zeros play a relevant role in various applications of the L-functions The paper begins with a survey of the classical material, presenting several applications to the class number problem and to the distribution of primes Recent results are then Preface VII outlined, dealing also with complex zeros on the critical line and with families of L-functions The last section deals with Linnik’s celebrated theorem on the least prime in an arithmetic progression, which uses many properties of the exceptional zero However, here the point of view is rather different from Linnik’s original approach In fact, a new proof of Linnik’s result based on sieve methods is given, with only a moderate use of L-functions Kaczorowski’s lectures present a survey of the axiomatic class S of Lfunctions introduced by Selberg Essentially, the main aim of the Selberg class theory is to prove that such an axiomatic class coincides with the class of automorphic L-functions Although the theory is rich in interesting conjectures, the focus of these lecture notes is mainly on unconditional results After a chapter on classical examples of L-functions and one on the basic theory, the notes present an account of the invariant theory for S The core of the theory begins with chapter 4, where the necessary material on hypergeometric functions is collected Such results are applied in the following chapters, thus obtaining information on the linear and non-linear twists which, in turn, yield a complete characterization of the degree functions and the non-existence of functions with degree between and 5/3 We are pleased to express our warmest thanks to the authors for accepting our invitation to the C.I.M.E session, and for agreeing to write the fine papers collected in this volume Alberto Perelli Carlo Viola Contents Producing Prime Numbers via Sieve Methods John B Friedlander “Classical” sieve methods Sieves with cancellation Primes of the form X + Y Asymptotic sieve for primes Conclusion References 18 28 38 47 47 Counting Rational Points on Algebraic Varieties D R Heath-Brown First lecture A survey of Diophantine equations 1.1 Introduction 1.2 Examples 1.3 The heuristic bounds 1.4 Curves 1.5 Surfaces 1.6 Higher dimensions Second lecture A survey of results 2.1 Early approaches 2.2 The method of Bombieri and Pila 2.3 Projective curves 2.4 Surfaces 2.5 A general result 2.6 Affine problems Third lecture Proof of Theorem 14 3.1 Singular points 3.2 The Implicit Function Theorem 3.3 Vanishing determinants of monomials 3.4 Completion of the proof Fourth lecture Rational points on projective surfaces 51 51 51 51 53 55 55 57 57 57 58 59 61 64 64 65 65 66 68 71 72 X Contents 4.1 Theorem – Plane sections 4.2 Theorem – Curves of degree or more 4.3 Theorem – Quadratic curves 4.4 Theorem – Large solutions 4.5 Theorem – Inequivalent representations 4.6 Theorem – Points on the surface E = Fifth lecture Affine varieties 5.1 Theorem 15 – The exponent set E 5.2 Completion of the proof of Theorem 15 5.3 Power-free values of polynomials Sixth lecture Sums of powers, and parameterizations 6.1 Theorem 13 – Equal sums of two powers 6.2 Parameterization by elliptic functions 6.3 Sums of three powers References 72 73 74 74 76 77 78 78 79 82 85 86 89 91 94 Conversations on the Exceptional Character Henryk Iwaniec 97 Introduction 97 The exceptional character and its zero 98 How was the class number problem solved? 101 How and why the central zeros work? 104 What if the GRH holds except for real zeros? 108 Subnormal gaps between critical zeros 109 Fifty percent is not enough! 112 Exceptional primes 114 The least prime in an arithmetic progression 117 9.1 Introduction 117 9.2 The case with an exceptional character 120 9.3 A parity-preserving sieve inequality 123 9.4 Estimation of ψX (x; q, a) 125 9.5 Conclusion 127 9.6 Appendix Character sums over triple-primes 128 References 130 Axiomatic Theory of L-Functions: the Selberg Class Jerzy Kaczorowski 133 Examples of L-functions 134 1.1 Riemann zeta-function and Dirichlet L-functions 134 1.2 Hecke L-functions 136 1.3 Artin L-functions 140 1.4 GL2 L-functions 145 1.5 Representation theory and general automorphic L-functions 155 The Selberg class: basic facts 159 2.1 Definitions and initial remarks 159 Contents XI 2.2 The simplest converse theorems 163 2.3 Euler product 166 2.4 Factorization 170 2.5 Selberg conjectures 174 Functional equation and invariants 177 3.1 Uniqueness of the functional equation 177 3.2 Transformation formulae 178 3.3 Invariants 181 Hypergeometric functions 186 4.1 Gauss hypergeometric function 186 4.2 Complete and incomplete Fox hypergeometric functions 187 4.3 The first special case: µ = 188 4.4 The second special case: µ > 191 Non-linear twists 193 5.1 Meromorphic continuation 193 5.2 Some consequences 196 Structure of the Selberg class: d = 197 6.1 The case of the extended Selberg class 197 6.2 The case of the Selberg class 200 Structure of the Selberg class: < d < 201 7.1 Basic identity 201 7.2 Fourier transform method 202 7.3 Rankin-Selberg convolution 204 7.4 Non existence of L-functions of degrees < d < 5/3 205 7.5 Dulcis in fundo 206 References 207 Axiomatic theory of L-functions: the Selberg class 201 Structure of the Selberg class: < d < 7.1 Basic identity Let F ∈ Sd# , < d < For positive α and sufficiently large σ we write ∞ F (s, α) = a(n) e(−nα) ns n=1 Moreover, let , A = (d − 1)qF−κ , d−1 d s∗ = κ s + − + iθF , σ ∗ = s∗ , κ= ∞ D(s, α) = a(n) n e A s n α n=1 κ With this notation we have the following result Theorem 7.1.1 ([23]) Let < d < 2, F ∈ Sd# , α > 0, and let J be an integer Then there exists a constant c0 = and polynomials Pj (s) with j J − and P0 (s) = c0 identically, such that for σ ∗ > σa (F ) J−1 ακ(ds−d/2+iθF +j) Pj (s)D(s∗ + jκ, α) + GJ (s, α), F (s, α) = qFκs (7.1) j=0 where GJ (s, α) is holomorphic for s in the half-plane σ ∗ > σa (F ) − κJ and continuous for α > Proof Let zN := + 2πiα, where N and α are real and positive, and let N ∞ FN (s, α) = a(n) −nzN e ns n=1 Using the inverse Mellin transform and the functional equation for F we have FN (s, α) = RN (s, α) + χ1 (s)F (s) − χ2 (s)zN F (s − 1) ∞ + ω Q1−2s n a(n) Hc ,s 1−s 2z n Q N n=1 where Hc (z, s) is the incomplete Fox hypergeometric function defined in Section 4.4, 202 Jerzy Kaczorowski χ1 (s) = and if c > χ2 (s) = if c < 0, if c > if c < 1, −w RN (s, α) = Res F (s + w)Γ (w)zN w=1−s Now we let N → ∞ and use Lemma 4.4.3 After some computations we prove our theorem For details see [23] Theorem 7.1.1 has some interesting immediate consequences Corollary 7.1.1 ([23]) Every F ∈ Sd# with < d < is entire Indeed, for < d < we have σ + d/2 − d−1 and hence σ ∗ > for σ > d/2 Hence the right hand side of (7.1) is holomorphic for σ > d/2 In particular, F (s) = F (s, 1) is holomorphic at s = σ ∗ = s∗ = We can now give a further short proof of Theorem 2.2.3 Corollary 7.1.2 ([23]) We have Sd# = ∅ for < d < Proof Let F (s) be an L-function in Sd# , with < d < We can assume without loss of generality that F (1) = Indeed, when F (1) = we replace F (s) by an appropriate shift F (s + iθ) Then F (s)ζ(s) belongs to Sd# , < d < 2, and has a pole at s = 1, a contradiction with Corollary 7.1.1 7.2 Fourier transform method Let X be a sufficiently large integer, ε > 0, ν, ρ positive constants with ρ > ν + 1, ω(y) ∈ C0∞ (R) with support contained in [−ν, ν] and such that ω(y) Let ω(x) be the Fourier transform of ω, and σε∗ = σa (F ) − κ − ε, sε = (d − 1)σε∗ − d + − iθF Moreover, let c1 = e ω1 −1/2 , η(y) = A + A y X g(s, y) = , κd x 1+ −κ d−1 f (x, n) = A n1/d − x1/d ω2 , , y0 = y λ 2X, y A d−1 κ(ds−d/2+iθF ) , n =A x x n = c1 x n σε∗ +κ/2 n x 1/d κ −1 , g(sε , y0 ) ω(y0 − ρ) |η (y0 )| , Axiomatic theory of L-functions: the Selberg class 2X Σ1 (x) = a(n) n=X x n σε∗ Σ2 (x) = ω(xκ − nκ ) e ρ(nκ − xκ ) , a(n) λ ω X n ω2 X 203 (7.2) n e f (x, n) x Lemma 7.2.1 ([23]) For every test function ω(y) there exists a shift ρ such that ω1 < ω2 and, for every ε > 0, Σ1 (x) = x−κ/2 Σ2 (x) + Oε X σa (F )−κ+ε as X → ∞, uniformly for X x 2X Proof For σ ∗ > σa (F ) we have, using Theorem 7.1.1 with J = 1, F (s, α) = c0 q κs ακ(ds−d/2+iθF ) D(s∗ , α) + h(s), where h(s) is holomorphic for σ > σa (F ) − κ We use periodicity in α We replace α in the above formula by α + and subtract both formulae After a suitable change of variables we arrive at the following equality (y > 0): ∞ ∞ a(n) a(n) κ e(nκ η(y)) + h1 (s∗, y), ∗ e(n y) = g(s, y) s s∗ n n n=1 n=1 (7.3) where h1 (s∗, y) is holomorphic for σ ∗ > σa (F ) − κ and continuous for y > Moreover, κ(ds−d/2+iθF ) y d−1 g(s, y) = + A and η(y) = A + A y d−1 −κ We now compute ∞ I := −∞ ∞ a(n) e(nκ y) e(−xκ y) dy s∗ n n=1 ω(y − ρ) (ρ > ν + 1) Integrating term by term we obtain ∞ I= a(n) ω(xκ − nκ ) e ρ(nκ − xκ ) s∗ n n=1 Using (7.3) we have ∞ I= a(n) ns∗ n=1 ∞ −∞ g(s, y) ω(y − ρ) e(nκ η(y) − xκ y) dy + E, 204 Jerzy Kaczorowski where E stands for a negligible error term Now we apply the saddle point method to evaluate integrals and the result follows after some computations (see [23] for details) Corollary 7.2.1 ([23]) Sd# = ∅ for < d < Proof We apply Lemma 7.2.1 We take x = n, an integer of size X (X is positive and large) We have Σ1 (n) = a(n) ω(0) + OK x−K for every positive K, by the fast decay of the Fourier transform and by κ > Indeed, we have (n + 1)κ − nκ nκ−1 X κ−1 and hence ω(k κ − nκ ) K X −K (k = n) Using Lemma 7.2.1 we therefore obtain a(n) X −κ/2 |a(n)| + X σa (F )−κ X σa (F )−κ/2+ε n∼X Consequently ∞ |a(n)| + σa (F ) − κ/2 If d < 3/2 then κ > 2, and we get a contradiction with the definition of the abscissa of absolute convergence Hence the corollary follows 7.3 Rankin-Selberg convolution Definition |a(n)|2 n−s F ⊗ F (s) = ( s > 2σa (F )) n Lemma 7.3.1 Let < d < and F ∈ Sd# Then F ⊗ F (s) is holomorphic for σ > σa (F ) − κ apart from a simple pole at s = Proof Our starting point is formula (7.3) which we rewrite for short as follows: L(s∗, y) = R(s∗, y) + h1 (s∗, y) (7.4) Let θ(y) be a positive, C ∞ function with compact support contained in (0, ∞) We compute the following convolution F(s∗ ) = ∞ −∞ θ(y)L(s∗, y)L(s∗, y) dy Axiomatic theory of L-functions: the Selberg class 205 We change the order of integration and summation Since κ > 1, only the diagonal terms matter here Thus after integrating term by term we obtain ∞ F(s∗ ) = F ⊗ F (2s∗ ) θ(y) dy + h(s∗ ), −∞ where h(s∗ ) is entire Using (7.4) we obtain ∞ F(s∗ ) = F ⊗ F (2s∗ ) θ(y)h(y)2s ∗ −1 −∞ ∗ dy + h2 (s∗ ), ∗ where h2 (s ) is holomorphic for σ > σa (F ) − κ Writing w = 2s∗ we have h3 (w) F ⊗ F (w) = h4 (w), where ∞ h3 (w) = −∞ θ(y)(1 − h(y))w−1 dy and h4 (w) is holomorphic for w > σa (F ) − κ Note that the entire function h3 (w) has a simple zero at w = 1, while by a suitable choice of θ(y) we can ensure that h3 (w) = for w ∈ [σa (F ) − κ, 2σa (F )] \ {1} Therefore the only possible pole of F ⊗ F (w) on the half-plane w > σa (F ) − κ is at w = Moreover w = is a simple pole of F ⊗ F (w), as follows from Landau’s theorem on Dirichlet series The lemma is proved Corollary 7.3.1 ([23]) For x → ∞ we have |a(n)|2 ∼ c0 X n x with a positive constant c0 7.4 Non existence of L-functions of degrees < d < 5/3 Theorem 7.4.1 ([23]) We have Sd# = ∅ for < d < Proof We consider the integral 2X |Σ1 (x)|2 e(x) dx, J(X) = X where X > is sufficiently large and Σ1 (x) is defined by (7.2) Moreover, let ∆ = X 1−κ+δ for a sufficiently small positive δ Owing to the decay of ω(x) and since κ > we have 2X n+∆ |a(n)|2 J(X) = (1 + O(∆)) n=X n−∆ x n 2σε∗ |ω(xκ − nκ )|2 dx + O X −M 206 Jerzy Kaczorowski for every M > It is easy to see that the integrals on the right hand side are X 1−κ Hence, using Corollary 7.3.1, we have J(X) X 2−κ (7.5) In order to obtain the upper bound for J(X) we apply Lemma 7.2.1, thus getting X −κ J(X) a(n) a(m) In,m + X 2σa (F )+1−2κ+ε , (7.6) ω1 X n,m ω2 X where 2X In,m = λ X If |n − m| n m e f (x, n) − f (x, m) + x dx λ x x X 2−κ then ∂ f (x, n) − f (x, m) + x ∂x |n − m| X κ−2 Hence, by the first derivative test, we get In,m for = |n − m| X 2−κ |n − m| X 2−κ , and In,n For |n − m| X 2−κ we have ∂2 f (x, n) − f (x, m) + x ∂x2 1/X, whence by the second derivative test we obtain In,m X 1/2 Inserting these estimates into (7.6) we obtain after some calculations J(X) X 7/2−κ Comparing (7.5) and (7.7) we obtain κ follows 3/2, i.e d (7.7) 5/3, and the result 7.5 Dulcis in fundo We end these lecture notes with the following converse theorems concerning the Riemann zeta function and the Dirichlet L-functions Formulations Axiomatic theory of L-functions: the Selberg class 207 are simple but proofs, although short, heavily depend on the main results of Chapters 5, and 7, and therefore are rather deep Theorem 7.5.1 ([24]) Let F ∈ Sd with d If the series ∞ aF (n) − ns n=1 converges for σ > 1/5 − δ with some δ > 0, then F (s) = ζ(s) Theorem 7.5.2 ([24]) Let F ∈ Sd with d If the 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Prachar, Primzahlverteilung, Springer-Verlag, Berlin-Heidelberg, 1978 ă [37] H.-E Richert, Uber Dirichletreihen mit Funktionalgleichung, Publ Inst Math Acad Serbe Sci (1957), 73-124 [38] A Selberg, Old and new conjectures and results about a class of Dirichlet series, Proc Amalfi Conference on Analytic Number Theory (Maiori, 1989), E Bombieri et al (eds.), Universit` a di Salerno, 1992, 367-385; Collected papers, Vol II, Springer-Verlag, Berlin-Heidelberg, 1991, 47-63 [39] J.-P Serre, Repr´esentations lin´eaires des groupes finis, Hermann, Paris, 1971 [40] F Shahidi, On non-vanishing of L-functions, Bull Amer Math Soc (1980), 462-464 [41] K Soundararajan, Strong multiplicity one for the Selberg class, Canad Math Bull 47 (2004), 468-474 [42] J Tate, Fourier analysis in number fields and Hecke’s zeta-functions, Thesis, Princeton, 1950; reproduced in: Algebraic number theory, J W S Cassels, A Fră ohlich (eds.), Academic Press, London, 1967 [43] E C Titchmarsh, The theory of the Riemann zeta function, 2nd ed., Clarendon Press, Oxford, 1988 [44] E C Titchmarsh, The theory of functions, 2nd ed., Oxford University Press, 1939 List of Participants Adhikari Sukumar Das Harish-Chandra R.I., India adhikari@mri.ernet.in Aliev Iskander Polish Academy of Science iskander@impan.gov.pl Avanzi Roberto Univ Essen mocenigo@exp-math uni-essen.de Basile Carmen Laura Imperial College, London laura.basile@ic.ac.uk Bourqui David Univ Grenoble bourqui@ujf-grenoble.fr Broberg Niklas Chalmers Univ nibro@math.chalmers.se Browning Timothy Math Inst., Oxford browning@maths.ox.ac.uk Bră udern Jă org Univ Stuttgart bruedern@mathematik uni-stuttgart.de Chamizo Fernando Univ Autonoma Madrid fernando.chamizo@uam.es 10 Chiera Francesco Univ di Padova chiera@math.unipd.it 11 Chinta Gautam Brown Univ chinta@math.brown.edu 12 Corvaja Pietro Univ di Udine corvaja@dimi.uniud.it 13 Dahari Arie Samuel Bar Ilan Univ dahari@math.biu.ac.il 14 De Roton Anne Univ de Bordeaux deroton@math.u-bordeaux.fr 15 Dvornicich Roberto Univ di Pisa dvornic@dm.unipi.it 16 Elsholtz Christian Tech Univ Clausthal elsholtz@math.tu-clausthal.de 17 Esposito Francesco Univ di Roma esposito@mat.uniroma1.it 18 Fischler St´ ephane Ec Normale Sup., Paris fischler@math.jussieu.fr 19 Friedlander John B Univ of Toronto frdlndr@math.toronto.edu (lecturer) 20 Garaev Moubariz Ac Sinica, Taiwan garaev@math.sinica.edu.tw 21 Heath-Brown D R Math Inst., Oxford rhb@maths.ox.ac.uk (lecturer) 22 Ivic Alexandar Serbian Acad Sc., Beograd aivic@rgf.bg.ac.yu 23 Iwaniec Henryk Rutgers Univ iwaniec@math.rutgers.edu (lecturer) 212 List of Participants 24 Kaczorowski Jerzy Adam Mickiewicz Univ., Pozna´ n kjerzy@amu.edu.pl (lecturer) 25 Kadiri Habiba Univ de Lille kadiri@agat.univ-lille1.fr 26 Kawada Koichi Iwate Univ kawada@iwate-u.ac.jp 27 Khemira Samy Univ Paris khemira@math.jussieu.fr 28 Languasco Alessandro Univ di Padova languasc@math.unipd.it 29 Laporta Maurizio Univ di Napoli laporta@matna2.dma.unina.it 30 Lau Yuk-Kam Hong Kong Univ yklau@maths.hku.hk 31 Longo Matteo Univ di Padova mlongo@math.unipd.it 32 Makatchev Maxim Univ of Pittsburg maxim@pitt.edu 33 Marcovecchio Raffaele Univ di Pisa marcovec@mail.dm.unipi.it 34 Marmi Stefano Univ di Udine e SNS marmi@sns.it 35 Melfi Giuseppe Univ Neuchatel Giuseppe.Melfi@unine.ch 36 Mitiaguine Anton Moscow State Univ mityagin@dnttm.ru 37 Molteni Giuseppe Univ di Milano giuseppe.molteni@mat.unimi.it 38 Ng Nathan Univ Montreal nathan@dms.umontreal.ca 39 Obukhovski Andrey Voronezh St Univ., Russia andrei@ob.vsu.ru 40 Pappalardi Francesco Univ Roma Tre pappa@mat.uniroma3.it 41 Perelli Alberto Univ di Genova perelli@dima.unige.it (editor) 42 Rocadas Luis UTAD, Portugal rocadas@utad.pt 43 Rodionova Irina Voronezh St Univ., Russia rodirina@hotmail.com 44 Schlickewei Hans Peter Univ Marburg hps@mathematik.uni-marburg.de 45 Skogman Howard SUNY Brockport hskogman@brockport.edu 46 Summerer Leonard ETH Zurich summerer@math.ethz.ch 47 Surroca Andrea Univ Paris surroca@math.jussieu.fr 48 Tchanga Maris Steklov Inst., Moscow maris changa@mail.ru 49 Traupe Martin mamt@math.tu-clausthal.de 50 Tubbs Robert Univ of Colorado, Boulder tubbs@euclid.colorado.edu 51 Ubis Adrian Univ Autonoma Madrid adrian.ubis@uam.es 52 Viola Carlo Univ di Pisa viola@dm.unipi.it (editor) 53 Viviani Filippo Univ di Roma viviani@mat.uniroma2.it 54 Vorotnikov Dmitry Voronezh St Univ., Russia mitvorot@mail.ru 55 Welter Michael Univ Kă oln mwelter@mi.uni-koeln.de 56 Zaccagnini Alessandro Univ di Parma zaccagni@math.unipr.it 57 Zannier Umberto SNS, Pisa u.zannier@sns.it 58 Zhang Qiao Columbia Univ., NY, USA qzhang@math.columbia.edu 59 Zudilin Wadim Lomonosov St Univ., Moscow wadim@ips.ras.ru LIST OF C.I.M.E SEMINARS Published by C.I.M.E 1954 Analisi funzionale Quadratura delle superficie e questioni connesse Equazioni differenziali non lineari 1955 Teorema di Riemann-Roch e questioni connesse Teoria dei numeri Topologia Teorie non linearizzate in elasticit` a, idrodinamica, aerodinamic Geometria proiettivo-differenziale 1956 Equazioni alle derivate parziali a caratteristiche reali 10 Propagazione delle onde elettromagnetiche 11 Teoria della funzioni di pi` u variabili complesse e delle funzioni automorfe 1957 12 Geometria aritmetica e algebrica (2 vol.) 13 Integrali singolari e questioni connesse 14 Teoria della turbolenza (2 vol.) 1958 15 Vedute e problemi attuali in relativit` a generale 16 Problemi di geometria differenziale in grande 17 Il principio di minimo e le sue applicazioni alle equazioni funzionali 1959 18 Induzione e statistica 19 Teoria algebrica dei meccanismi automatici (2 vol.) 20 Gruppi, anelli di Lie e teoria della coomologia 1960 21 Sistemi dinamici e teoremi ergodici 22 Forme differenziali e loro integrali 1961 23 Geometria del calcolo delle variazioni (2 vol.) 24 Teoria delle distribuzioni 25 Onde superficiali 1962 26 Topologia differenziale 27 Autovalori e autosoluzioni 28 Magnetofluidodinamica 1963 29 Equazioni differenziali astratte 30 Funzioni e variet` a complesse 31 Propriet` a di media e teoremi di confronto in Fisica Matematica 1964 32 33 34 35 1965 36 Non-linear continuum theories 37 Some aspects of ring theory 38 Mathematical optimization in economics Relativit` a generale Dinamica dei gas rarefatti Alcune questioni di analisi numerica Equazioni differenziali non lineari Published by Ed Cremonese, Firenze 1966 39 40 41 42 Calculus of variations Economia matematica Classi caratteristiche e questioni connesse Some aspects of diffusion theory 1967 43 Modern questions of celestial mechanics 44 Numerical analysis of partial differential equations 45 Geometry of homogeneous bounded domains 1968 46 Controllability and observability 47 Pseudo-differential operators 48 Aspects of mathematical logic 1969 49 Potential theory 50 Non-linear continuum theories in mechanics and physics and their applications 51 Questions of algebraic varieties 1970 52 53 54 55 1971 56 Stereodynamics 57 Constructive aspects of functional analysis (2 vol.) 58 Categories and commutative algebra 1972 59 Non-linear mechanics 60 Finite geometric structures and their applications 61 Geometric measure theory and minimal surfaces 1973 62 Complex analysis 63 New variational techniques in mathematical physics 64 Spectral analysis 1974 65 Stability problems 66 Singularities of analytic spaces 67 Eigenvalues of non linear problems 1975 68 Theoretical computer sciences 69 Model theory and applications 70 Differential operators and manifolds Relativistic fluid dynamics Theory of group representations and Fourier analysis Functional equations and inequalities Problems in non-linear analysis Published by Ed Liguori, Napoli 1976 71 Statistical Mechanics 72 Hyperbolicity 73 Differential topology 1977 74 Materials with memory 75 Pseudodifferential operators with applications 76 Algebraic surfaces Published by Ed Liguori, Napoli & Birkhă auser 1978 77 Stochastic differential equations 78 Dynamical systems 1979 79 Recursion theory and computational complexity 80 Mathematics of biology 1980 81 Wave propagation 82 Harmonic analysis and group representations 83 Matroid theory and its applications Published by Springer-Verlag 1981 84 Kinetic Theories and the Boltzmann Equation 85 Algebraic Threefolds 86 Nonlinear Filtering and Stochastic Control (LNM 1048) (LNM 947) (LNM 972) 1982 87 Invariant Theory 88 Thermodynamics and Constitutive Equations 89 Fluid Dynamics (LNM 996) (LNP 228) (LNM 1047) 1983 90 Complete Intersections 91 Bifurcation Theory and Applications 92 Numerical Methods in Fluid Dynamics (LNM 1092) (LNM 1057) (LNM 1127) 1984 93 Harmonic Mappings and Minimal Immersions 94 Schră odinger Operators 95 Buildings and the Geometry of Diagrams (LNM 1161) (LNM 1159) (LNM 1181) 1985 96 Probability and Analysis 97 Some Problems in Nonlinear Diffusion 98 Theory of Moduli (LNM 1206) (LNM 1224) (LNM 1337) 1986 99 Inverse Problems 100 Mathematical Economics 101 Combinatorial Optimization (LNM 1225) (LNM 1330) (LNM 1403) 1987 102 Relativistic Fluid Dynamics 103 Topics in Calculus of Variations (LNM 1385) (LNM 1365) 1988 104 Logic and Computer Science 105 Global Geometry and Mathematical Physics (LNM 1429) (LNM 1451) 1989 106 Methods of nonconvex analysis 107 Microlocal Analysis and Applications (LNM 1446) (LNM 1495) 1990 108 Geometric Topology: Recent Developments 109 H∞ Control Theory 110 Mathematical Modelling of Industrial Processes (LNM 1504) (LNM 1496) (LNM 1521) 1991 111 Topological Methods for Ordinary Differential Equations 112 Arithmetic Algebraic Geometry 113 Transition to Chaos in Classical and Quantum Mechanics (LNM 1537) (LNM 1553) (LNM 1589) 1992 114 Dirichlet Forms 115 D-Modules, Representation Theory, and Quantum Groups 116 Nonequilibrium Problems in Many-Particle Systems (LNM 1563) (LNM 1565) (LNM 1551) 1993 117 Integrable Systems and Quantum Groups 118 Algebraic Cycles and Hodge Theory 119 Phase Transitions and Hysteresis (LNM 1620) (LNM 1594) (LNM 1584) 1994 120 Recent Mathematical Methods in Nonlinear Wave Propagation 121 Dynamical Systems 122 Transcendental Methods in Algebraic Geometry (LNM 1640) (LNM 1609) (LNM 1646) 1995 123 Probabilistic Models for Nonlinear PDE’s 124 Viscosity Solutions and Applications 125 Vector Bundles on Curves New Directions (LNM 1627) (LNM 1660) (LNM 1649) 1996 126 Integral Geometry, Radon Transforms and Complex Analysis 127 Calculus of Variations and Geometric Evolution Problems 128 Financial Mathematics (LNM 1684) (LNM 1713) (LNM 1656) 1997 129 Mathematics Inspired by Biology 130 Advanced Numerical Approximation of Nonlinear Hyperbolic Equations 131 Arithmetic Theory of Elliptic Curves 132 Quantum Cohomology (LNM 1714) (LNM 1697) (LNM 1716) (LNM 1776) 1998 133 134 135 136 137 Optimal Shape Design Dynamical Systems and Small Divisors Mathematical Problems in Semiconductor Physics Stochastic PDE’s and Kolmogorov Equations in Infinite Dimension Filtration in Porous Media and Industrial Applications (LNM (LNM (LNM (LNM (LNM 1999 138 139 140 141 142 Computational Mathematics driven by Industrial Applications Iwahori-Hecke Algebras and Representation Theory Hamiltonian Dynamics - Theory and Applications Global Theory of Minimal Surfaces in Flat Spaces Direct and Inverse Methods in Solving Nonlinear Evolution Equations (LNM 1739) (LNM 1804) (LNM 1861) (LNM 1775) (LNP 632) 2000 143 144 145 146 147 Dynamical Systems Diophantine Approximation Mathematical Aspects of Evolving Interfaces Mathematical Methods for Protein Structure Noncommutative Geometry (LNM 1822) (LNM 1819) (LNM 1812) (LNCS 2666) (LNM 1831) 2001 148 149 150 151 Topological Fluid Mechanics Spatial Stochastic Processes Optimal Transportation and Applications Multiscale Problems and Methods in Numerical Simulations to appear (LNM 1802) (LNM 1813) (LNM 1825) 2002 152 Real Methods in Complex and CR Geometry 153 Analytic Number Theory 154 Imaging (LNM 1848) (LNM 1891) to appear 2003 155 156 157 158 (LNM 1856) to appear to appear (LNM 1871) 2004 159 Representation Theory and Complex Analysis 160 Nonlinear and Optimal Control Theory 161 Stochastic Geometry 2005 162 Enumerative Invariants in Algebraic Geometry and String Theory to appear 163 Calculus of Variations and Non-linear Partial Differential Equations to appear 164 SPDE in Hydrodynamics: Recent Progress and Prospects to appear 2006 165 Pseudo-Differential Operators, Quantization and Signals 166 Mixed Finite Elements, Compatibility Conditions, and Applications 167 From a Microscopic to a Macroscopic Description of Complex Systems 168 Quantum Transport: Modelling, Analysis and Asymptotics Stochastic Methods in Finance Hyperbolic Systems of Balance Laws Symplectic 4-Manifolds and Algebraic Surfaces Mathematical Foundation of Turbulent Viscous Flows 1740) 1784) 1823) 1715) 1734) to appear to appear (LNM 1892) announced announced announced announced ... order of summation we obtain S(A, z) = an n µ(d) = = µ(d) d|P (z) an n d|(n,P (z)) µ(d) d|n d|P (z) an = n≡0 (mod d) µ(d)Ad (x), d|P (z) say This is just (a more general version of) the Legendre... us return to the general version of the Legendre formula, namely S(A, z) = µ(d)Ad (x) d|P (z) = A(x) µ(d)g(d) + d|P (z) µ(d)rd (x) d|P (z) Recall that this was based on the basic property: µ(d)... − g(p) + small, + λ+ d g(d) = F (s) d p|P (z) − λ− d g(d) = F (s) d − g(p) + small, p|P (z) 20 John B Friedlander where the functions F ± are given by the following diagram Fκ+ (s) s= Fκ− (s)